Asolid sphere of uniform density starts from rest and rolls without slipping down an inclined plane with angle θ = 30°. the sphere has mass m = 8 kg and radius r = 0.19 m . the coefficient of static friction between the sphere and the plane is μ = 0.64. what is the magnitude of the frictional force on the sphere?

The space between two 15-in.-long concentric cylinders is filled with glycerin (viscosity = 8.5 × 10-3 lb·s/ft2). the inner cylinder has a radius of 1 in. and the gap width between cylinders is 0.1 in. determine (a) the torque and (b) the power required to rotate the inner cylinder at 180 rev/min. the outer cylinder is fixed. assume the velocity distribution in the gap to be linear.